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A Hamiltonian reduction approach is defined, studied, and finally used toderive asymptotic models of internal wave propagation in density stratifiedfluids in two-dimensional domains. Beginning with the general Hamiltonianformalism of Benjamin [1] for an ideal, stably stratified Euler fluid, thecorresponding structure is systematically reduced to the setup of twohomogeneous fluids under gravity, separated by an interface and confinedbetween two infinite horizontal plates. A long-wave, small-amplitudeasymptotics is then used to obtain a simplified model that encapsulates most ofthe known properties of the dynamics of such systems, such as bidirectionalwave propagation and maximal amplitude travelling waves in the form of fronts.Further reductions, and in particular devising an asymptotic extension ofDirac's theory of Hamiltonian constraints, lead to the completely integrableevolution equations previously considered in the literature for limiting formsof the dynamics of stratified fluids. To assess the performance of theasymptotic models, special solutions are studied and compared with those of theparent equations.

Simple two-layer dispersive models in the Hamiltonian reduction formalism